Identical particles Masatsugu Sei Suzuki Department of Physics, SUNY at Binghamton (Date: March 04, 2017)
In classical mechanics, identical particles (such as electrons) do not lose their "individuality", despite the identity of their physical properties. In quantum mechanics, the situation is entirely different. Because of the Heisenberg's principle of uncertainty, the concept of the path of an electron ceases to have any meaning. Thus, there is in principle no possibility of separately following each of a number of similar particles and thereby distinguishing them. In quantum mechanics, identical particles entirely lose their individuality. The principle of the in-distinguishability of similar particles plays a fundamental part in the quantum theory of system composed of identical particles. Here we start to consider a system of only two particles. Because of the identity of the particles, the states of the system obtained from each other by interchanging the two particles must be completely equivalent physically.
1 System of particles 1 and 2
k' , k"
k' k" k" k' a 1 2 b 1 2
Even though the two particles are indistinguishable, mathematically a and b are distinct kets for
k' k" . In fact we have a b 0 . Suppose we make a measurement on the two particle system.
k' : state of one particle and k" : state of the other particle.
We do not know a priori whether the state ket is k' k" or k" k' or -for that matter- any a 1 2 b 1 2 linear combination of the two: ca a cb b .
2 Exchange degeneracy A specification of the eigenvalue of a complete set of observables does not completely determine the state ket.
Mathematics of permutation symmetry:
Pˆ Pˆ k' k" k" k' . 12 a 12 1 2 1 2 b
Clearly
ˆ ˆ ˆ 2 P21 P12 , and P12 1.
ˆ Under P12 , particle 1 having k' becomes particle 1 having k" ; particle 2 having k" becomes particle 1 having k' . In other words, it has the effect of interchanging 1 and 2.
((Note))
Identical particles 1 9/3/2017 Matrix element of Pˆ Pˆ in terms of k' k" and k" k' 21 12 a 1 2 b 1 2
Pˆ Pˆ k' k" k" k' , 12 a 12 1 2 1 2 b
Pˆ Pˆ k" k' k' k" . 12 b 12 1 2 1 2 a
ˆ ˆ Matrix element of P21 P12
ˆ 0 1 P12 1 0 or
a b
a 0 1
b 1 0
ˆ Eigensystem[ P12 ]
= 1 (symmetric):
P12 symm symm
1 1 ( k' k" k" k' ) (1ˆ Pˆ ) k' k" symm 2 1 2 1 2 2 12 1 2 where the symmetrizer is defined by
1 Sˆ (1ˆ Pˆ ) 2 12
= -1 (anti-symmetric)
P12 anti anti
1 1 ( k' k" k" k' ) (1ˆ Pˆ ) k' k" anti 2 1 2 1 2 2 12 1 2 where the antisymmetrizer is defined by
1 Aˆ (1ˆ Pˆ ) 2 12
Identical particles 2 9/3/2017 Our consideration can be extended to a system made up of many identical particles. A transposition is a permutation which simply exchange the role of two of the particles, without touching others.
Pˆ k' k" ..... k i k i1 ..... k j ..... k' k" ..... k j k i1 ..... k i ..... ij 1 2 i i1 j 1 2 i i1 j
ˆ ˆ ˆ The transposition operators Pij are Hermitian ( Pij Pij )
ˆ 2 ˆ Pij 1
ˆ ˆ So that Pij is an unitary operator. The allowed eigenvalues of Pij are ±1. It is important to note, however, that in general
ˆ ˆ [Pij , Pkl ] 0.
(i) The first example
ˆ Now we consider a permutation operator P123 for
Pˆ k' k" k"' k' k" k"' k"' k' k" 123 1 2 3 2 3 1 1 2 3 for the system of 3 identical particles ( k' k" k''' ) 1 2 3
123 P 123 231
This means replacement of 12, 2 3, 3 1.
123 123213 P P P 123 231 213231 12 13
Quantum mechanically this is not correct. The correct one is
ˆ ˆ ˆ P123 P13 P12
(ii) The second example
123 12 3132 P P P 123 231 132231 23 12 or quantum mechanically
ˆ ˆ ˆ P123 P12P23
(iii) The third example
Identical particles 3 9/3/2017
123 123321 P P P 132 312 321312 13 12 or quantum mechanically
ˆ ˆ ˆ P132 P12 P13
((Proof))
Pˆ Pˆ k' k" k''' Pˆ k''' k" k' k" k''' k' 12 13 1 2 3 12 1 2 3 1 2 3 and
Pˆ k' k" k''' k' k" k''' k" k''' k' 132 1 2 3 3 1 2 1 2 3
ˆ ˆ ˆ Therefore we have P132 P12 P13 .
Any permutation operators can be broken into a product of transposition operators.
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 P132 P23 P12 P13 P23 P12 P13 P23 P12 P23 ...
The decomposition is not unique. However, for a given permutation, it can be shown that the parity of the number of transposition into which it can be broken down is always the same: it is called the parity of the permutation.
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 P132 P23P12 P13P23 P12P13 P23P12P23 even permutation
ˆ ˆ ˆ ˆ ˆ P123 P12 P23 P13 P12 : even permutation
ˆ P23 odd permutation
ˆ P12 odd permutation
ˆ P31 odd permutation
((Note))
ˆ ˆ ˆ P132 P321 P213
3 Symmetrizer and antisymmetrizer Now we consider the two Hermitian operators
ˆ 1 ˆ S P : symmetrizer N!
Identical particles 4 9/3/2017 ˆ 1 ˆ A P : antisymmetrizer N!
ˆ ˆ where =1 if P is an even permutation and = -1 if P is an odd permutation.
Sˆ Sˆ and Aˆ Aˆ .
((Theorem-1)) ˆ If P 0 is an arbitrary permutation operator, we have
ˆ ˆ ˆ ˆ ˆ P 0 S SP 0 S
ˆ ˆ ˆ ˆ P 0 A AP 0 0 A (1)
((Proof)) This is due to the fact that
ˆ ˆ ˆ P 0P P such that
0 or
2 0 0
Consequently
ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ P 0S P 0P P S N! N!
ˆ ˆ 1 ˆ ˆ 1 ˆ ˆ P 0 A P 0 P 0 P 0 A N! N!
From Eq.(1), we see the following theorem
((Theorem-2))
Sˆ 2 Sˆ
Aˆ 2 Aˆ and
AˆSˆ SˆAˆ 0 Identical particles 5 9/3/2017
((Proof))
ˆ 2 1 ˆ ˆ 1 ˆ ˆ S P S S S N! N!
ˆ 2 1 ˆ ˆ 1 2 ˆ ˆ A P A A A N! N!
ˆ ˆ ˆ 1 ˆ ˆ 1 ˆ AS A P S S 0 N! N! since half the are equal to 1 and half the equal to -1. We also note that
1 2 1 1 1 N! N!
Sˆ and Aˆ are therefore projectors. Their action on any ket of the state space yields a completely symmetric or completely antisymmetric ket.
ˆ ˆ ˆ P 0S S
Pˆ Aˆ Aˆ 0 0
ˆ S S
ˆ A A
((Example))
For N = 3,
1 Sˆ [1ˆ Pˆ Pˆ Pˆ Pˆ Pˆ ] 6 12 23 31 123 132 and
1 Aˆ [1ˆ Pˆ Pˆ Pˆ Pˆ Pˆ ] 6 12 23 31 123 132 where
ˆ ˆ ˆ ˆ ˆ ˆ P123 P12P23 , P132 P12P13
Identical particles 6 9/3/2017 1 Sˆ Aˆ (1ˆ Pˆ Pˆ ) 1ˆ 3 123 132
4 Symmetrization postulate The system containing N identical particles are either totally symmetrical under the interchange of any pair (boson), or totally antisymmetrical under the interchange of any pair (fermion).
ˆ Pij N ,B N ,B ,
ˆ Pij N ,F N ,F ,
where N ,B is the eigenket of N identical boson systems and N ,F is the eigenket of N identical fermion systems.
((Note)) It is an empirical fact that a mixed symmetry does not occur.
Even more remarkable is that there is a connection between the spin of a particle and the statistics obeyed by it:
Half-integer spin particles are fermion, while integer-spin particles are bosons.
5 Pauli exclusion principle Wolfgang Ernst Pauli (April 25, 1900 – December 15, 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or Pauli principle," involving spin theory, underpinning the structure of matter and the whole of chemistry.
http://en.wikipedia.org/wiki/Wolfgang_Pauli
Electron is a fermion. No two electrons can occupy the same state. We discuss the framatic difference between fermions and bosons. Let us consider two particles. Each of which can occupy only two states k' and k" . For a system of two fermions, we have no choice
Identical particles 7 9/3/2017
1 1 k' k' [ k' k" k' k" ] 1 2 1 2 2 1 k'' k" 2 2 1 2
For bosons, there are three states.
k' k' , 1 2
k" k" , 1 2
1 ( k' k" k" k' ) . 2 1 2 1 2
In contrast, for “classical particles” satisfying Maxwell-Boltzmann (M-B) statitics with no restriction on symmetry, we have altogether four independent states.
k' k" , k" k' , k' k' and k" k" 1 2 1 2 1 2 1 2
We see that in the fermion case, it is impossible for both particles to occupy the same state.
6 Transformation of observables by permutation For simplicity, we consider a specific case where the two particle state ket is completely specified by the eigenvalues of a single observable Aˆ for each of the particle.
Aˆ a' a" a' a' a" 1 1 2 1 2 and
Aˆ a' a" a" a' a" 2 1 2 1 2
Since
Pˆ Aˆ Pˆ 1Pˆ a' a" Pˆ Aˆ Pˆ 1 a" a' 12 1 12 12 1 2 12 1 12 1 2 or
Pˆ Aˆ Pˆ 1Pˆ a' a" Pˆ Aˆ a' a" 12 1 12 12 1 2 12 1 1 2 a' Pˆ a' a" 12 1 2 a' a" a' 1 2 Aˆ a" a' 2 1 2 Aˆ Pˆ a' a" 2 12 1 2 we obtain Identical particles 8 9/3/2017
ˆ ˆ ˆ 1 ˆ P12 A1 P12 A2 .
Similarly, we have
ˆ ˆ ˆ 1 ˆ P21 A2 P21 A1 .
ˆ It follows that P12 must change the particle label of observables.
ˆ ˆ ˆ ˆ There are also observables, such as A1 B2 , A1 B2 , which involve both indices simultaneously.
ˆ ˆ ˆ ˆ 1 ˆ ˆ P12 (A1 B2 )P12 A2 B1
ˆ ˆ ˆ ˆ 1 ˆ ˆ ˆ 1 ˆ ˆ ˆ 1 ˆ ˆ P12 A1 B2 P12 P12 A1 P12 P12 B2 P12 A2 B1
These results can be generalized to all observables which can be expressed in terms of observables which can ˆ ˆ ˆ be expressed in terms of observables of the type of A1 and B2 , to be denoted by O(1,2) .
ˆ ˆ ˆ 1 ˆ P12 O(1, 2 )P12 O(2,1) where Oˆ (2,1) is the observable obtained from Oˆ (1,2) by exchanging indices 1 and 2 throughout.
ˆ Os (1,2) is said to be symmetric if
ˆ ˆ Os (1, 2) Os (2,1) or
ˆ ˆ [Os (1,2),P12 ] 0
Symbolic observables commute with the permutation operator.
ˆ In general. the observables Os (1,2,3,..., N) which are completely symmetric under exchange of indices 1, 2, ..., N commute with all the transposition operators, and with all the permutation operators
7 Example Let us now consider a Hamiltonian of a system of two identical particles.
1 1 Hˆ pˆ 2 pˆ 2 V ( rˆ rˆ ) V (rˆ ) V (rˆ ) 2m 1 2m 2 pair 1 2 ext 1 ext 2
Clearly we have
ˆ ˆ ˆ 1 ˆ P12 H P12 H or
Identical particles 9 9/3/2017
ˆ ˆ [P12, ,H ] 0
ˆ ˆ 2 ˆ P12 is a constant of the motion. Since P12 1, the eigenvalue of P12 allowed are ±1.
Hˆ E
ˆ P12
ˆ 2 ˆ 2 P12 P12 or
= ±1.
It therefore follows that if the two-particle state ket is symmteric (antisymmettric) to start with, it remains so at all times.
(i) N = 2 case We can define the symmetrizer and antisymmetrtizer as follows.
1 1 Sˆ (1 Pˆ ) Aˆ (1 Pˆ ) 2 12 2 12
Sˆ Aˆ 1ˆ
1 1 1 1 Sˆ 2 (1ˆ Pˆ ) (1ˆ Pˆ ) (1ˆ 2Pˆ 1ˆ) (1ˆ Pˆ ) Sˆ 2 12 2 12 4 12 2 12
1 1 1 1 Aˆ 2 (1ˆ Pˆ ) (1ˆ Pˆ ) (1ˆ 2Pˆ 1ˆ) (1ˆ Pˆ ) Aˆ 2 12 2 12 4 12 2 12
Then we have
1 S ( k' k" k" k' ) 2 1 2 1 2 and
1 A ( k' k" k" k' ) 2 1 2 1 2 where
1 1 Sˆ k' k" (1 Pˆ ) k' k" ( k' k" k" k' ) 1 2 2 12 1 2 2 1 2 1 2 Identical particles 10 9/3/2017
1 1 Aˆ k' k" (1 Pˆ ) k' k" ( k' k" k" k' ) 1 2 2 12 1 2 2 1 2 1 2
(ii) N = 3 Cases
1 Sˆ (1 Pˆ Pˆ Pˆ Pˆ Pˆ ) 6 12 23 31 123 132
1 Aˆ (1 Pˆ Pˆ Pˆ Pˆ Pˆ ) 6 12 23 31 123 132
1 Sˆ Aˆ (1 Pˆ Pˆ ) 1 3 123 132
k' k" k''' 1 1 1 1 Aˆ k' k" k''' k' k" k''' 1 2 3 3! 2 2 2 k' k" k''' 3 3 3
Slater determinant Aˆ k' k" k''' is zero if two of individual states coincide. We obtain Pauli’s exclusion principle. 1 2 3
8 Generalized method (Tomonaga) We now consider a system consisting of many spins.
ˆ ˆ ˆ ˆ ˆ S S1 S2 S3 ... SN
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ S S (S1 S2 S3 ... SN ) (S1 S2 S3 ... SN ) or
N 2 N 1 ˆ 2 ˆ 2 ˆ ˆ ˆ 2 2 ˆ ˆ S Sn 2(Sn Sn' ) σn (σn σn' ) n1 nn' 4 n1 2 nn' or
Sˆ 2 3N 1 ˆ ˆ ˆ 2 1 (σn σn' ) 4 2 nn'
Here we define an operator
ˆ 2 ˆ O Pnn' N(N 1) nn'
(i) Oˆ is Hermitian Identical particles 11 9/3/2017
((Proof))
ˆ 2 ˆ 2 ˆ ˆ O Pnn' Pnn' O N(N 1) nn' N(N 1) nn'
(ii)
[Pˆ,Oˆ] 0.
As shown in the Appendix,
PˆPˆ Pˆ Pˆ nn' rr rn'
where the pair (n, n') corresponds to the pair (rn , rn' ) in one-to one.
ˆ ˆ ˆ ˆ P Pnn' Pnn'P nn. nn.
______We assume that
1 Pˆ (1ˆ σˆ σˆ ) (Dirac exchange interaction) nn' 2 n n'
2 1 2 1 N(N 1) 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ O (1 σn σn' ) [ 1 σn σn' ] N(N 1) nn' 2 N(N 1) 2 2 2 nn' or
1 2 ˆ ˆ ˆ ˆ O [1 σn σn' ] 2 N(N 1) nn'
Using the relation,
Sˆ 2 3N 1 ˆ ˆ ˆ 2 1 (σn σn' ) , 4 2 nn' we get
ˆ 1 ˆ 2 2 ˆ 2 3N 1 N 4 ˆ 4 1 ˆ 2 O [1 ( 2 S )] [ 1 2 S ] 2 N(N 1) 2 2 (N 1) N(N 1)
ˆ 2 ˆ ˆ 2 2 ˆ [S ,O] 0 . When the eigenvalue of S is given by S(S 1 ) , the eigenvalue of O is equal to
Identical particles 12 9/3/2017 1 N 4 4S(S 1) [ ] 2 N 1 N(N 1)
The eigenvalue of Oˆ , () , specifies the symmetry.
Oˆ S S leading to the eigenvalue 1 for the symmetric state
Oˆ A A leading to the eigenvalue 1 for the antisymmetric state.
(i) For N = 2,
D1/2 x D1/2 = D1 + D0
1 [2 2S(S 1)] 1 S(S 1) 2
When S = 1, = 1 (symmetric). When S = 0, = -1 (anti-symmetric).
(ii) For N = 3,
D1/2 x D1/2 x D1/2 = D3/2 + 2 D1/2
1 1 2 [ S(S 1)] 2 2 3
When S = 3/2, = 1 (symmetric). When S = 1/2, = 0.
(1) j = 3/2 (symmetric)
3 3 , (1) (2) (3) 2 2
3 1 (1) (2) (3) (1)(2)(3) (1) (2)(3) , 2 2 3
3 1 (1)(2)(3) (1) (2)(3) (1)(2) (3) , 2 2 3
3 3 , (1)(2)(3) 2 2
______(ii) j = 1/2.
Identical particles 13 9/3/2017 1 1 (1) (2) (3) (1)(2) (3) 2 (1) (2)(3) , 2 2 6
1 1 (1)(2)(3) 2(1)(2) (3) (1) (2)(3) , 2 2 6
______(iii) j = 1/2
1 1 (1) (2) (3) (1)(2) (3) [(1) (2)) (1)(2)] (3) , 2 2 2 2
1 1 (1) (2)(3) (1)(2)(3) [(1) (2) (1)(2)](3) , 2 2 2 2
(iii) For N = 4,
D1/2 x D1/2 x D1/2 x D1/2 = D2 + 3D1 + 2D0
S(S 1) 6
For S = 2, = 1 (symmetric). For S = 1, = 1/3. For S = 0, = 0. (antisymmetric)
9. Summary: fermion and boson The Hamiltonian H of the system (in the absence of a magnetic field) does not contain the spin operators, and hence, when it is applied to the wave function, it has no effect on the spin variables. The wave function of the system of particles can be written in the form of product,
space spin ,
where space depends only on the coordinate of the particles and spin only on their spins. space spin should be anti-symmetric since electrons are Fermions. The system containing N identical particles are either totally symmetrical under the interchange of any pair (boson), or totally antisymmetrical under the interchange of any pair (fermion).
ˆ Pij N ,B N ,B ,
ˆ Pij N ,F N ,F ,
where N ,B is the eigenket of N identical boson systems and N ,F is the eigenket of N identical fermion systms. Identical particles 14 9/3/2017
((Note)) It is an empirical fact that a mixed symmetry does not occur.
Even more remarkable is that there is a connection between the spin of a particle and the statistics obeyed by it:
Half-integer spin particles are fermion, obeying the Fermi-Dirac statistic, while integer-spin particles are bosons, obeying the Bose-Einstein statistics.
10. Ground state for systems with many electron We construct the form of wavefunctions for the system with two, three, four, and five electrons (identical particles, fermion) using the Slater determinant. We assume that each electron has spin states
, .
We also assume the orbital state: a n 1 b n 2 , c n 3 .., where n is the principal quantum number. It is reasonable to consider that the ground state can be expressed by electrons occupied by
a , a taking into account of the Pauli’s exclusion principle. For the system with three electrons, the ground state can be expressed by
a , a , b
For the system with the four electrons, the ground state can be expressed by
a , a , b , b
For the system with the five electrons, the ground state can be expressed by
a , a , b , b , ` c
(a) Two particles state For the two states,
a a , and a a we have the Slater determinant
1 a11 a11 1 a1a2 (12 12 ) 2 a22 a22 2
Identical particles 15 9/3/2017 where index 1, 2 denotes the particle 1 and particle 2. Clearly, the state vector is antisymmetric under the exchange of two particles.
(b) Three particles state For the three states,
a a , a a , and b b we have the Slater determinant
a11 a11 b11 1 a a b 6 2 2 2 2 2 2 a33 a33 b33 1 [a a ( )b a a ( )b a a ( )b ] 6 2 3 2 3 2 3 1 1 1 3 1 3 1 3 2 2 1 2 1 2 1 2 3 3
This state vector is antisymmetric under the exchange between any two particles (such that 1 2 ).
(c) Four particle state For the four states,
a a , a a , b b , and b b we have the Slater determinant
a11 a11 b11 b11 1 a a b b 2 2 2 2 2 2 2 2 24 a33 a33 b33 b33
a4 4 a44 b44 b4 4
b1b2 (12 1 2 )a3a4 (34 34 )
b2b3 (23 23 )a1a4 (14 1 4 )
b2b4 (24 2 4 )a1a4 (14 14 )
b1b3 (13 13 )a2a4 (24 24 )
b1b4 (14 14 )a2a3 ( 23 23 )
b3b4 (34 34 )a1a2 (12 1 2 )
This state vector is antisymmetric under the exchange between any two particles (such that 1 2 ), but symmetric under the two types exchange (such that 1 2 and 3 4 )
(d) Five particle state For the five states Identical particles 16 9/3/2017
a a , a a , b b , b b , c c we have the Slater determinant
a11 a11 b11 b11 c11
a22 a22 b22 b22 c22 1 a a b b c 120 3 3 3 3 3 3 3 3 3 3 a44 a44 b44 b44 c44
a55 a55 b55 b55 c55 1 [a a ( )b b ( )c ...] 120 1 2 1 2 1 2 3 4 3 4 4 3 5 5
((Mathematica))
Identical particles 17 9/3/2017 a1 1 a1 1 Clear "Global` " ;A1 ; a2 2a2 2 Det A1 FullSimplify a1 a2 2 1 1 2
a1 1a11b11 B1 a2 2a22b22 ; a3 3a33b33 Det B1 FullSimplify a1 a3 b2 a2 b3 2 3 1 a2 a3 b1 a1 b3 1 3 2 a3 a2 b1 a1 b2 1 2 3
a1 1a11b11b11 a2 2a22b22b22 C1 ; a3 3a33b33b33 a4 4a44b44b44 Det C1 FullSimplify b2 a3 a4 b1 2 1 1 2 4 3 3 4 a1 a4 b3 3 2 2 3 4 1 1 4 a3 b4 3 1 1 3 4 2 2 4 a2 a4 b1 b3 3 1 1 3 4 2 2 4 b4 a3 b1 3 2 2 3 4 1 1 4
a1 b3 2 1 1 2 4 3 3 4
______REFERENCES S. Tomonaga Angular momentum and spin (Misuzo Syobo, Tokyo, 1989) [in Japanese]. J.J. Sakurai Modern Quantum Mechanics, Revised Edition (Addison-Wesley, Reading Massachusetts, 1994). L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon Press, Oxford, 1977). John S. Townsend , A Modern Approach to Quantum Mechanics, second edition (University Science Books, 2012). David H. McIntyre, Quantum Mechanics A Paradigms Approach (Pearson Education, Inc., 2012). C. Cohen-Tannoudji, B. Du, and F. Laloe, Quantum Mechanics vol. 2 (John Wiley & Sons, 1977).
APPENDIX-I ((Townsend textbook))
Identical particles 18 9/3/2017 For a symmetric spatial state under the exchange of two particles, we have
1 dr' dr"Sˆ r',r" r',r" Sˆ dr' dr" r',r" r',r" S 2 S S and for an anti-symmetric spatial state under the exchange of two particles, we have
1 dr' dr"Aˆ r',r" r',r" Aˆ dr' dr" r',r" r',r" A 2 A A where
r',r" r' r" 1 2
1 1 Sˆ (1ˆ Pˆ ) , Aˆ (1ˆ Pˆ ) 2 12 2 12
((Proof))
1 dr' dr"Sˆ r',r" r',r" Sˆ S 2 S 1 1 dr' dr" (1ˆ Pˆ ) r',r" r',r" (1ˆ Pˆ ) 2 2 12 12 S 1 dr' dr"(1ˆ Pˆ ) r',r" r',r" (1ˆ Pˆ ) 4 12 12 S 1 dr' dr"( r',r" r",r' r',r" 2 S 1 dr' dr"( r',r" r',r" r",r' r',r" ) 2 S S 1 dr' dr"( r',r" r',r" r",r' r",r' ) 2 S S dr' dr"( r',r" r',r" S where we use the relation
ˆ r',r" S r',r" P12 S r",r' S .
Similarly,
Identical particles 19 9/3/2017 1 dr' dr"Aˆ r',r" r',r" Aˆ A 2 A 1 dr' dr"(1ˆ Pˆ ) r',r" r',r" (1ˆ Pˆ ) 4 12 12 A 1 dr' dr"( r',r" r",r' r',r" A 2 1 dr' dr"( r',r" r',r" r",r' r',r" ) 2 A A 1 dr' dr"( r',r" r',r" r",r' r",r' ) 2 A A dr' dr"( r',r" r',r" A where we use the relation
ˆ r',r" S r",r' P12 S r",r' S
ˆ r',r" A r',r" P12 A r",r' A
ˆ ˆ P12 S S , P12 A A
______APPENDIX-II Property of permutation operator
ˆ 2 ˆ 1. P12 1
((Proof))
Pˆ 2 ' Pˆ ' ' 12 1 2 12 1 2 1 2
ˆ 2 ˆ ˆ leading to the relation P12 1. The operator P12 is its own inverse.
ˆ ˆ ˆ 2. P12 is Hermitian operator: P12 P12
((Proof))
' Pˆ ' ( ' )( ' ) 1 2 12 1 2 1 2 1 2 , ' ',
From the definition
Identical particles 20 9/3/2017 ' Pˆ ' ' Pˆ ' * 1 2 12 1 2 1 2 12 1 2 [( ' )( ' )]* 1 2 1 2 * ( , ' ', )
, ' ',
ˆ ˆ This leads to the relation, P12 P12 .
ˆ ˆ ˆ ˆ ˆ ˆ 3. P12 is unitary operator: P12 P12 P12P12 1
((Proof))
ˆ ˆ ˆ ˆ ˆ P12 P12 P12P12 1
ˆ ˆ ˆ ˆ ˆ P12P12 P12P12 1
______APPENDIX-III Permutation operator (a)
1 2 3 4 P' 4 3 2 1 1 2 3 42 1 3 4 2 1 3 44 3 2 1 1 2 3 41 2 3 4 2 1 3 43 4 2 1
P12P
1 2 3 43 4 2 1 P' 3 4 2 14 3 2 1 1 2 3 41 2 3 4 3 4 2 11 2 4 3
PP34 leading to the expression in quantum mechanics;
ˆ ˆ ˆ ˆ PP12 P34P
(b)
Identical particles 21 9/3/2017 1 2 3 4 P' 3 1 4 2 1 2 3 43 2 1 4 3 2 1 43 1 4 2 1 2 3 41 2 3 4 3 2 1 44 1 3 2
P13P
1 2 3 4 P' 3 1 4 2 1 2 3 44 1 3 2 4 1 3 23 1 4 2 1 2 3 41 2 3 4 4 1 3 21 2 4 3
PP34 leading to the expression in quantum mechanics;
ˆ ˆ ˆ ˆ PP13 P34P
Identical particles 22 9/3/2017